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Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
Symmetric tensors and symmetric tensor rank
 Scientific Computing and Computational Mathematics (SCCM
, 2006
"... Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. An ..."
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Abstract. A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank1 orderk tensor is the outer product of k nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank1 tensors, each of them being symmetric or not. The rank of a symmetric tensor is the minimal number of rank1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases, and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz, is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most r is not closed, unless r = 1. Key words. Tensors, multiway arrays, outer product decomposition, symmetric outer product decomposition, candecomp, parafac, tensor rank, symmetric rank, symmetric tensor rank, generic symmetric rank, maximal symmetric rank, quantics AMS subject classifications. 15A03, 15A21, 15A72, 15A69, 15A18 1. Introduction. We
SYMMETRIC TENSOR DECOMPOSITION
, 2009
"... We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented ..."
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Cited by 30 (5 self)
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We present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is twofold. First it permits an efficient computation of the decomposition of any tensor of subgeneric rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank.
Generic and typical ranks of multiway arrays
 Linear Algebra Appl
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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Cited by 27 (5 self)
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
SUBTRACTING A BEST RANK1 APPROXIMATION MAY INCREASE TENSOR RANK
"... Is has been shown that a best rankR approximation of an orderk tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computi ..."
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Is has been shown that a best rankR approximation of an orderk tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank1 approximations. The reason for this is that subtracting a best rank1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for realvalued 2 × 2 × 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2 × 2 × 2 tensors (which have rank 2 or 3), subtracting a best rank1 approximation will result in a tensor that has rank 3 and lies on the boundary between the rank2 and rank3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank1 approximation has increased the tensor rank.
Lowrank decomposition of multiway arrays: A signal processing perspective
 In IEEE SAM
, 2004
"... In many signal processing applications of linear algebra tools, the signal part of a postulated model lies in a socalled signal subspace, while the parameters of interest are in onetoone correspondence with a certain basis of this subspace. The signal subspace can often be reliably estimated from ..."
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In many signal processing applications of linear algebra tools, the signal part of a postulated model lies in a socalled signal subspace, while the parameters of interest are in onetoone correspondence with a certain basis of this subspace. The signal subspace can often be reliably estimated from measured data, but the particular basis of interest cannot be identified without additional problemspecific structure. This is a manifestation of rotational indeterminacy, i.e., nonuniqueness of lowrank matrix decomposition. The situation is very different for three or higherway arrays, i.e., arrays indexed by three or more independent variables, for which lowrank decomposition is unique under mild conditions. This has fundamental implications for DSP problems which deal with such data. This paper provides a brief tour of the basic elements of this theory, along with many examples of application in problems of current interest in the signal processing community. Keywords: Threeway analysis, lowrank decomposition, parallel factor analysis (PARAFAC), canonical decomposition (CAN
Generic and typical ranks of threeway arrays
 Research Report ISRN I3S/RR200629FR, I3S, SophiaAntipolis
"... The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a giv ..."
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The concept of tensor rank, introduced in the twenties, has been popularized at the beginning of the seventies. This has allowed to carry out Factor Analysis on arrays with more than two indices. The generic rank may be seen as an upper bound to the number of factors that can be extracted from a given tensor. We explain in this short paper how to obtain numerically the generic rank of tensors of arbitrary dimensions, and compare it with the rare algebraic results already known at order three. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, or tensors with free entries. Résumé La notion de rang tensoriel, proposée dans les années vingt, a été popularisée au début des années soixantedix. Ceci a permis de mettre en oeuvre l’Analyse de Facteurs sur des tableaux de données comportant plus de deux indices. Le rang générique peut être vu comme une borne supérieure sur le nombre de facteurs pouvant être extraits d’un tenseur donné. Nous expliquons dans ce court article comment trouver numériquement le rang générique d’un tenseur de dimensions arbitraires, et le comparons aux quelques rares résultats algébriques déjà connus à l’ordre trois. Nous examinons notamment les cas des tenseurs symétriques, des tenseurs à tranches matricielles symétriques, ou des tenseurs à éléments libres.
A comparison of different notions of ranks of symmetric tensors
, 2012
"... We introduce various notions of rank for a high order symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smooth able rank. Weanalyze the stratification induced by these ranks. The mutual relations between these ..."
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Cited by 5 (1 self)
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We introduce various notions of rank for a high order symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smooth able rank. Weanalyze the stratification induced by these ranks. The mutual relations between these stratifications, allowus to describethe hierarchyamongall the ranks. We show that strict inequalities are possible between rank, border rank, extension rank and catalecticant rank. Moreover we show that scheme length, generalized rank and extension rank coincide.
DOI: 10.1007/1133600612782 SUFFICIENT CONDITIONS FOR UNIQUENESS IN CANDECOMP/PARAFAC AND INDSCAL WITH RANDOM COMPONENT MATRICES
, 2006
"... A key feature of the analysis of threeway arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case wher ..."
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A key feature of the analysis of threeway arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the threeway array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal.